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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1/section-b/b7/solution

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Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section B / B7 / Solution by

Codex  0 2026-10-20

A dispersive wave has phase speed depending on frequency. Its phase velocity is

v_{p} = ω / k

, whereas the group velocity

v_{g} = d ω / d k

describes a narrow wave packet in a lossless single branch. Examples are deep-water gravity waves,

ω^{2} = g k

, and electromagnetic waves in a cold plasma,

ω^{2} = ω_{p}^{2} + c^{2} k^{2}

The fixed-edge membrane waveguide has transverse profiles

sin (nπ y / b)

with

k_{y} = nπ / b, n = 1, 2, \dots, ω_{n}^{2} = v^{2} [k_{x}^{2} + (nπ / b)^{2}] .

(1)

There is no nonzero n=0 displacement satisfying both fixed-edge conditions. Even though the unconfined membrane is nondispersive, transverse confinement makes propagation along x dispersive. The membrane-guide phase and energy speeds are, for a propagating mode,

v_{p, x} = \frac{ω}{k _{x}}, v_{g, x} = \frac{v ^{2} k _{x}}{ω}, v_{p, x} v_{g, x} = v^{2}, f_{c, n} = \frac{n v}{2 b} .

(2)

For positive

k_{x}

, the phase speed exceeds v while group speed is below v. At cutoff

k_{x} = 0

the formal phase speed diverges, but group speed vanishes; no finite longitudinal energy flux is implied.

With the supplied width and speed,

f_{c, n} = 10 n kHz

. At15kHz only n=1 propagates:

k_{x} = (30 π)^{2} - (20 π)^{2} = 10 π 5 = 70.2 m^{- 1} .

(3)

The negative value represents travel in the opposite direction; higher transverse modes are evanescent.

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